where \(\mathrm {Cov}_p\)  denotes the covariance under \(\phi _p\) .
For \(q=1\) , the sum of covariances gets nicely rewritten in terms of pivotal edges, i.e. edges which, when switched from close to open, change the occurrence of the event. In particular, it is possible to prove that for crossing events of a rectangle of size \(R\)  and edges that are far from the boundary of the rectangle, the probability of being pivotal is of the order of the probability \(\pi _4(p,R)\)  that the two extremities of a given edge \(e\)  belong to different clusters of radius at least \(R\)  (when \(p=p_c\) , we simply write \(\pi _4(R)\) ).
This property was used crucially in [1]} and ultimately leads to Kesten's scaling relation \(L(p)^2 \pi _4(L(p)) \asymp (p-p_c)^{-1}\) .
The description in terms of pivotal edges is wrong for random-cluster models with \(q>1\)  as the covariance between an edge and crossing events at scale \(R\)  is no longer of the order of \(\pi _4(p,R)\) .
